Characterizing Face and Flag Vector Pairs for Polytopes.

Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs (f i , f j) of face numbers of 4-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers (f 0 , f 03) for 4-polytopes. Furthermore, we describe the pairs of face numbers (f 0 , f d - 1) for d-polytopes; this description is complete for even d ≥ 6 except for finitely many exceptional pairs that are "small" in a well-defined sense, while for odd d we show that there are also "large" exceptional pairs. Our proofs rely on the insight that "small" pairs need to be defined and to be treated separately; in the 4-dimensional case, these may be characterized with the help of the characterizations of the 4-polytopes with at most eight vertices by Altshuler and Steinberg (1984). [ABSTRACT FROM AUTHOR]